Placeholder Substructures II: Meta-Fractals, Made of Box-Kites, Fill Infinite-Dimensional Skies
Abstract
Description
Zero-divisors (ZDs) derived by Cayley-Dickson Process (CDP) from N-dimensional hypercomplex numbers (N a power of 2, at least 4) can represent singularities and, as N approaches infinite, fractals -- and thereby,scale-free networks. Any integer greater than 8 and not a power of 2 generates a meta-fractal or "Sky" when it is interpreted as the "strut constant" (S) of an ensemble of octahedral vertex figures called "Box-Kites" (the fundamental building blocks of ZDs). Remarkably simple bit-manipulation rules or "recipes" provide tools for transforming one fractal genus into others within the context of Wolfram's Class 4 complexity.
31 pp. Second of 3-part "theorem/proof" exposition of 78-slide Powerpoint from Wolfram Science's NKS 2006, available at http://wolframscience.com/conference/2006/presentations/materials/demarrais.ppt [v2: small fixes][v3: Added new Appendix B and small number of corrections (pp. 7, 14, 20) RE: 2nd type of box-kite flow pattern.]
31 pp. Second of 3-part "theorem/proof" exposition of 78-slide Powerpoint from Wolfram Science's NKS 2006, available at http://wolframscience.com/conference/2006/presentations/materials/demarrais.ppt [v2: small fixes][v3: Added new Appendix B and small number of corrections (pp. 7, 14, 20) RE: 2nd type of box-kite flow pattern.]